Scalar Quasi-Normal Modes in Black Hole Gravitational Lensing

TL;DR

The paper investigates how quasi-normal modes (QNMs) are excited when a black hole gravitationally lenses scalar field perturbations in Schwarzschild spacetime. It develops a time-domain, multipole (mode-sum) method to propagate the scalar field, validating the approach against high-resolution 3D simulations. A key finding is that a burst can non-resonantly excite a broad spectrum of high-$l$ QNMs, with late-time signals well described by QNM templates, while the lensed waveforms form a highly directional Gaussian beam whose amplitude remains nearly constant in the near field and whose oscillations largely cancel due to multi-mode interference. These results offer a clean, linear-perturbation pathway to understand BH lensing signals and motivate extensions to gravitational waves, where tensorial QNMs are expected to follow similar qualitative behavior.

Abstract

We investigate the excitation of quasi-normal modes (QNMs) in gravitational lensing by a Schwarzschild black hole using a scalar field model. By employing a time-domain mode-sum method, we analyze the complex interplay between an incident burst signal and the black hole spacetime. We find that the incident waves can non-resonantly excite a substantial number of high-$l$ modes, with amplitudes for modes as high as l=20 remaining significant compared to the fundamental l=0 mode. We confirm through QNM template fitting that the late-time behaviors of these excited modes are indeed QNMs. After passing through the black hole, we find that the lensed waves form a highly directional and coherent Gaussian beam whose cross-sectional intensity profile is well-described by a Gaussian profile. Unlike spherical waves, this beam's amplitude does not decrease with distance from the black hole but remains nearly constant in the near-field region. Moreover, due to the superposition of numerous QNMs, oscillations largely cancel each other out. The lensed temporal waves do not exhibit typical oscillatory patterns.

Figures (10)

• Figure 1: Comparison of a direct 3D numerical simulation and the time-domain mode-sum method. The left column displays snapshots of the wave signals in the $x$-$z$ plane at different times from the direct high-resolution 3D simulation. The middle column shows the corresponding results reconstructed using our time-domain mode-sum method. The right column presents the residuals (the difference) between the two methods. Overall, the time-domain mode-sum method exhibits excellent agreement with the 3D simulation, even in cases of complex wave dynamics. The 3D simulations are in isotropic coordinates of the Schwarzschild BH.
• Figure 2: Temporal waveforms observed on the $x$-axis. The left panel shows waveforms in the forward direction at $x/M = 20,\,60$, while the right panel shows waveforms in the backward direction at $x/M = -20,\,-60$. Waveforms from the direct 3D simulation (solid lines) are compared with those reconstructed using the time-domain mode-sum method (dashed lines) up to $l=63$. The results from the two methods are in excellent agreement.
• Figure 3: Comparison of the solutions for $u_l$ between different numerical methods across the first three non-zero modes $l=0,\,1,\,3$. Each row corresponds to a different mode and each column is observed at different radial distances of $\rho/M =2,\, 30, \,40$. The black solid lines show results from our FEM in isotropic coordinates. The blue dashed lines show results from a Finite Difference Method (FDM) in tortoise coordinates with the inner absorbing boundary set at $r_*/M=-40$ (equivalent to $\rho / M \approx 0.50003$). In the middle column, the yellow dash-dotted lines are the best-fit QNM templates overlaid on $u_l$, fitted over the time intervals indicated by the shaded regions.
• Figure 4: Snapshots of the 3D numerical results in the $x$-$z$ plane obtained from the time-domain mode-sum method. The solid black disk represents the location of the BH, with its radius indicating the event horizon. The color bar to the right shows the amplitude of these wave signals. Each panel illustrates the lensed wave signals at different times. The 3D simulations are in isotropic coordinates of the Schwarzschild BH.
• Figure 5: Temporal waveforms observed on the $x$-axis at $x/M=10,\,30$ in forward direction (upper-panel) and $x/M=-10,\,-30$ in backward direction (lower-panel).